Question

A 5-card hand is drawn from a deck of standard playing cards.

Hints: 1. A deck of standard playing cards consists of 52 cards 2. Each deck contains 13 different kinds of ranks, ranging from 1 (Ace) to 13 (King) 3. Each rank contains 4 suits: spade, club, heart, and diamond 4. To summary, each deck has 13 cards in each suit and 4 cards in each rank

1. If we only consider one suit, how many 5-card hands have NO cards with the same rank?

2. How many 5-card hands in which all the cards have different ranks are there?

3. How many 5-card hands have at least two cards with the same rank?

Answer 1

A 5-card hand is drawn from a deck of standard playing cards. The number of 5-card hands with no cards with the same rank is 154,440. The number of 5-card hands with all cards having different ranks is also 154,440. The number of 5-card hands with at least two cards with the same rank is 111,920.

Step-by-step explanation:

In a 5-card hand drawn from a deck of standard playing cards, there are different scenarios to consider:

1. If we only consider one suit, there are 13 ranks to choose from. For the first card, we can choose any of the 13 ranks. For the second card, we can choose from the remaining 12 ranks, and so on. So, the total number of 5-card hands with no cards of the same rank is 13 * 12 * 11 * 10 * 9 = 154,440.
   
2. If we want all the cards to have different ranks, we can still consider one suit. For the first card, we can choose from any of the 13 ranks. For the second card, we can choose from the remaining 12 ranks, and so on. So, the total number of 5-card hands with all cards of different ranks is 13 * 12 * 11 * 10 * 9 = 154,440.
   
3. To calculate the number of 5-card hands with at least two cards of the same rank, we need to consider different scenarios. We can have 2 cards of the same rank and 3 cards of different ranks, or 3 cards of the same rank and 2 cards of different ranks, or 4 cards of the same rank and 1 card of a different rank, or all 5 cards of the same rank.
   Let's calculate each scenario:

• 2 cards of the same rank and 3 cards of different ranks: There are 13 options for the rank of the 2 cards, and 12 options for the rank of the remaining 3 cards. So, the total number of 5-card hands in this scenario is (13 * 12) * (4 * 3 * 2) = 3744.
• 3 cards of the same rank and 2 cards of different ranks: There are 13 options for the rank of the 3 cards, and 12 options for the rank of the remaining 2 cards. So, the total number of 5-card hands in this scenario is (13) * (4 * 3 * 2) * (12 * 11) = 72,576.
• 4 cards of the same rank and 1 card of a different rank: There are 13 options for the rank of the 4 cards, and 12 options for the rank of the remaining 1 card. So, the total number of 5-card hands in this scenario is (13) * (4) * (12 * 11 * 10) = 34,560.
• All 5 cards of the same rank: There are 13 options for the rank of the 5 cards. So, the total number of 5-card hands in this scenario is 13 * (4 * 3 * 2 * 1) = 1,040.

In total, the number of 5-card hands with at least two cards of the same rank is 3744 + 72,576 + 34,560 + 1,040 = 111,920.